#115375

order	512
nilpotency class	3
isoclinism	isoclinic class of #111483
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#60949	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

837 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,9)(3,10)(4,6,14,8)(7,13)(11,15)(12,16)(17,18,21,26,22,25,34,39)(19,23,29,36,28,37,43,48)(20,30,32,40,31,24,33,38)(27,35,42,46,41,45,44,47);(2,6)(3,11)(4,14)(8,9)(12,13)(15,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,35)(25,37)(26,36)(30,45)(31,41)(32,42)(33,44)(34,43)(38,46)(39,48)(40,47);(1,3)(2,7)(4,12)(5,13)(6,15)(8,16)(9,10)(11,14)(17,20)(18,24)(19,27)(21,33)(22,31)(23,35)(25,30)(26,40)(28,41)(29,44)(32,34)(36,47)(37,45)(38,39)(42,43)(46,48)

Polycyclic presentation:

2^9|a^2=g,d^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,d^c=di,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22571);